Tunneling corrections VTST

The integrals extend from — so and +So lying at the edges of the barrier. For the truncated parabola the solution reduces to Equation 6.8. Note that the tunneling corrections introduced in Sections 6.3.1 and 6.3.1.1 can be used either with TST or with VTST. 

Lasaga and Gibbs (1991) investigated kinetic isotope effects ( CH4/ CH4) of Equation (71) using TST and the Eckart tunneling correction (Johnston 1966). The predicted KIE values, in general, overestimate all the experimental values except that of Davidson et al. (1987). Related to this study, Xiao (unpublished results) performed a steepest descent calculation from the transition state (Fig. 9). These calculations of the minimum energy path are preliminaries needed for VTST calculations. 

The AG value deduced from the PMF is corrected by replacing classical vibrational partition functions by their quantum homolog. Recrossing, tunnelling and non-classical reflection effects can be included in the transmission coefficient by various procedures. This ensemble-average variational transition state theory with multidimensional tunnelling (EA-VTST/MT) method was applied to proton and hydride transfers in various enzymes such as yeast enolase, liver alcohol dehydrogenase and triosephosphate isomerase. For a review, see ref. 3 and the chapter by J. Gao in this book. 

In Fig. 5.1, thermal rate constants obtained from accurate full-dimensional calculations are compared to transition state theory results [40] (left panel) and reduced dimensionality quantum calculations (right panel). Experimental results [41] are also displayed. Classical transition state theory (TST) and variational transition state theory (VTST) drastically underestimate the thermal rate constant. This shows that tunneling effects are very prominent in the H2 + OH reaction. Including different types of tunneling corrections [40], increased values of the thermal rate constant can be obtained. However, it should be noted that different tunneling corrections result in considerably different results. This reflects the ambiguity of quantum transition state... 

Figure 6.19 Comparison between the rate constants of the reactions H+Hj and H+Dj, in units of mol dm sec , calculated by the scISM with Eckart-barrier tunnelling corrections and by the VTST with least-action tunnelling corrections (dotted lines) with the experimental rates.

In our own applications of VTST to bimolecular reactions we have also used the "quantum mechanical" formulation of section II.A in which internal states are quantized (quasiclassically) and tunneling effects are included (quantum mechanically or semiclassically). We have compared the rate constants calculated by our quantized formulation, with and without tunneling corrections, to accurate quantum mechanical equilibrium rate constants for the same assumed potential energy surfaces for several collinear reactions and one three-dimensional reaction.This work, reviewed elsewhere,shows that ... 

Abstract Some of the successes and several of the inadequacies of transition state theory (TST) as applied to kinetic isotope effects are briefly discussed. Corrections for quantum mechanical tunneling are introduced. The bulk of the chapter, however, deals with the more sophisticated approach known as variational transition state theory (VTST). 

In both TST and VTST, quantum mechanical tunneling is introduced into the rate constant expression as a correction factor usually referred to as k. A short discussion of k which is used largely with TST is presented in Section 6.3.1. Tunneling has been explored much more thoroughly in connection with VTST and this work will be discussed later. 

There are two corrections to equation (12) that one might want to make. The first has to do with dynamical factors [19,20] i.e., trajectories leave Ra, crossing the surface 5/3, but then immediately return to Ra. Such a trajectory contributes to the transition probability Wfia, but is not really a reaction. We can correct for this as in variational transition-state theory (VTST) by shifting Sajj along the surface normals. [8,9] The second correction is for some quantum effects. Equation (14) indicates one way to include them. We can simply replace the classical partition functions by their quantum mechanical counterparts. This does not correct for tunneling and interference effects, however. 

Melissas and Truhlar (1993a) studied the kinetic isotope effects (CD4/CH4) of Equation (71) using TST, CVT, and interpolated VTST (IVTST), which uses the small curvature tunneling (SCT) correction (Melissas and Truhlar 1993b). Their calculations show that accuracy of the KIE prediction increased dramatically from TST to IVTST. 

Tunneling occurs when a configuration, that has an energy lower than an energy barrier, nonetheless surmounts it due to quantum mechanical effects. In such cases, adjustments of the rate constant due to tunneling become necessary to obtain improved accuracy. These corrections in TST and VTST are in the form of a correction coefficient K such that... 

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